Propagation of chaos in infinite horizon and numerical stability for stochastic McKean-Vlasov equations

This paper focuses on the numerical stability of stochastic McKean-Vlasov equations (SMVEs) via the stochastic particle method. Firstly, the long-time propagation of chaos in the mean-square sense is obtained, and the almost sure propagation in infinite horizon is also proved. Next, when the coefficients satisfy linear growth conditions, the mean-square and almost sure exponential stabilities of the Euler-Maruyama (EM) scheme associated with the corresponding interacting particle system are shown through an ingenious manipulation of empirical measure. Then, for the case that the state variables in drift and diffusion are both superlinear, the mean-square exponential stability of the backward EM scheme for the interacting system is achieved without the particle corruption, which is a novel conclusion. Moreover, under the linear growth condition on the diffusion coefficient, the almost sure stability of the backward EM scheme is studied. Combining these assertions enables the numerical solutions to reproduce the stabilities of the original SMVEs. The examples, including a feedback control problem and a stochastic opinion dynamics model, are provided to demonstrate the importance of theoretical analysis of numerical stability.